Упр.1109 ГДЗ Алимов 10-11 класс (Алгебра)

1) С 1/4 + C2/4 + C3/4 + C4/4;
2) C1/5 + C2/5+C3/5 + C4/5;
3) C0/7 + C7/7 + C1/7 + C6/7 + C2/7 + C5/7+ C3/7+ C4/7;
4) C 0/6 + C 6/6 + C 1/6 + C 5/6 + C 2/6 + C 4/6 + C 3/6;
5) C2/12 + C3/12 + C4/13 + C5/14;
6) C3/9 + C4/9 + C5/10+ C6/11;
7) C 7/21 — C 7/20;
8) C9/25- C9/24;
9) C 10/15 — C 9/14;
10) C8/13 — C7/12.

1. $$C_4^1+C_4^2+C_4^3+C_4^4=\left(C_4^0+C_4^1+C_4^2+C_4^3+C_4^4\right)-C_4^0=2^4-1=15.$$
2. $$C_5^1+C_5^2+C_5^3+C_5^4=\left(C_5^0+C_5^1+C_5^2+C_5^3+C_5^4+C_5^5\right)-C_5^0-C_5^5=2^5-1-1=30.$$
3. $$C_7^0+C_7^7+C_7^1+C_7^6+C_7^2+C_7^5+C_7^3+C_7^4=2^7=128.$$
4. $$C_6^0+C_6^6+C_6^1+C_6^5+C_6^2+C_6^4+C_6^3=2^6=64.$$
5. $$C_{12}^2+C_{12}^3+C_{12}^4+C_{12}^5=C_{13}^3+C_{13}^4+C_{13}^5=C_{14}^4+C_{14}^5=C_{15}^5.$$
   $$C_{15}^5=\frac{15!}{5!\,10!}=\frac{15\cdot14\cdot13\cdot12\cdot11}{5\cdot4\cdot3\cdot2}=3003.$$
6. $$C_9^3+C_9^4+C_9^5+C_9^6=C_{10}^4+C_{10}^5+C_{10}^6=C_{11}^5+C_{11}^6=C_{12}^6.$$
   $$C_{12}^6=\frac{12!}{6!\,6!}=\frac{12\cdot11\cdot10\cdot9\cdot8\cdot7}{6\cdot5\cdot4\cdot3\cdot2}=924.$$
7. $$C_{21}^7-C_{20}^7=C_{20}^6.$$
   $$C_{20}^6=\frac{20!}{6!\,14!}=\frac{20\cdot19\cdot18\cdot17\cdot16\cdot15}{6\cdot5\cdot4\cdot3\cdot2}=38760.$$
8. $$C_{25}^9-C_{24}^9=C_{24}^8.$$
   $$C_{24}^8=\frac{24!}{8!\,16!}=\frac{24\cdot23\cdot22\cdot21\cdot20\cdot19\cdot18\cdot17}{8\cdot7\cdot6\cdot5\cdot4\cdot3\cdot2}=735471.$$
9. $$C_{15}^{10}-C_{14}^9=C_{14}^{10}.$$
   $$C_{14}^{10}=C_{14}^4=\frac{14!}{4!\,10!}=\frac{14\cdot13\cdot12\cdot11}{4\cdot3\cdot2}=1001.$$
10. $$C_{13}^8-C_{12}^7=C_{12}^8.$$
    $$C_{12}^8=C_{12}^4=\frac{12!}{4!\,8!}=\frac{12\cdot11\cdot10\cdot9}{4\cdot3\cdot2}=495.$$

Ответ

1) 15; 2) 30; 3) 128; 4) 64; 5) 3003; 6) 924; 7) 38760; 8) 735471; 9) 1001; 10) 495.